 InfinitesimalPseudoGroupNormalizer - Maple Help

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GroupActions[InfinitesimalPseudoGroupNormalizer] - find the normalizer of a finite dimensional Lie algebra of vector fields in an (infinite-dimensional) pseudo-Lie algebra of vector fields

Calling Sequences

InfinitesimalPseudoGroupNormalizer(Gamma, options)

Parameters

Gamma                 - a list, a basis for a Lie algebra of vector fields on a manifold $M$

options               - (optional) arguments that can be given in any order and are described as follows

ansatz                - ansatz= Z, where Z is a vector field on $M$

unknowns              - unknowns = U, where U is a list of the unknown functions appearing in the vector field $Z$

auxiliaryequations    - auxiliaryequations = E, where E is a list of additional partial differential equations to be imposed on the unknowns functions U

parameters            - parameters = P, where P is a set of parameters appearing in the vector fields $\mathrm{Γ}$

liealgebra                         -  liealgebra  = name, where name is the string or name for the abstract Lie algebra defined by $\mathrm{Γ}$

output                - output = "general", "pde"  or "list"

other options         - optional arguments to be passed to pdsolve Description

 • Let   be a -dimensional Lie algebra of vector fields, defined on a manifold Let be an infinite dimensional Lie algebra of vector fields on $M$, whose general element depends upon a certain number of arbitrary functions and suppose  Then the normalizer of in  is

Norfor all = { for  and where the coefficients are constant}.

The vector fields always belong to Nor(. The procedure InfinitesimalPseudoGroupNormalizer uses the pdsolve command to calculate Nor(modulo the vector fields in $\mathrm{Γ}$ but including vector fields in the center of $\mathrm{Γ}$.

 • With the calling sequence InfinitesimalPseudoGroupNormalizer(Gamma), the normalizer of $\mathrm{Γ}$ in the full infinitesimal pseudo-group of all vector fields on $M$ is computed. With output = "general" (the default value), a single vector field, depending upon arbitrary constants and functions, is returned. If the output depends only on constants (that is, Nor( is a finite-dimensional algebra) and output = "list", then a list of vectors is returned. In this case the vector fields in the center of are removed. With output = "pde", the determining differential equations for the normalizer are returned.
 • With the keyword argument ansatz = Z, the procedure InfinitesimalPseudoGroupNormalizer calculates Nor(, where is the infinitesimal pseudo-group defined by the vector field Z. With this keyword option, the unknown functions in Z must be explicitly listed using the keyword argument unknowns = U. Additional algebraic and differential constraints on the unknown functions in Z may be specified with the keyword argument auxiliaryequations = E. Note that the full system of differential equations for Z is likely to be inconsistent if the vector fields in do not satisfy the differential constraints defined by E.
 • If the abstract Lie algebra determined by has been calculated and initialized (see LieAlgebraData, DGsetup), then this information can be passed to InfinitesimalPseudoGroupNormalizer with the keyword argument liealgebra = name.
 • The keyword argument parameters = P will invoke the case-splitting capabilities of pdsolve . In this case, the output of InfinitesimalPseudoGroupNormalizer will be a sequence of normalizers for both the generic and special values of the parameters.
 • The command InfinitesimalPseudoGroupNormalizer is part of the DifferentialGeometry:-GroupActions package.  It can be used in the form InfinitesimalPseudoGroupNormalizer(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-InfinitesimalPseudoGroupNormalizer(...). Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{GroupActions}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$$\mathrm{with}\left(\mathrm{Library}\right):$

Example 1.

First define a 1-dimensional manifold with coordinate $\left[x\right]$.

 > $\mathrm{DGsetup}\left(\left[x\right],M\right):$

On $M$, define 1 and 2-dimensional Lie algebras of vector fields respectively.

 M > $\mathrm{Γ11}≔\mathrm{evalDG}\left(\left[\mathrm{D_x}\right]\right)$
 ${\mathrm{Γ11}}{:=}\left[{\mathrm{D_x}}\right]$ (2.1)
 M > $\mathrm{Γ12}≔\mathrm{evalDG}\left(\left[\mathrm{D_x},x\mathrm{D_x}\right]\right)$
 ${\mathrm{Γ12}}{:=}\left[{\mathrm{D_x}}{,}{x}{}{\mathrm{D_x}}\right]$ (2.2)

Find the normalizer of these two Lie algebras in the Lie algebras of all vector fields on $M$.

 M > $\mathrm{N11}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{Γ11}\right)$
 ${\mathrm{N11}}{:=}\left({\mathrm{_C1}}{}{x}{+}{\mathrm{_C2}}\right){}{\mathrm{D_x}}$ (2.3)
 M > $\mathrm{N11list}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{Γ11},\mathrm{output}="list"\right)$
 ${\mathrm{N11list}}{:=}\left[{x}{}{\mathrm{D_x}}\right]$ (2.4)
 M > $\mathrm{N12}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{Γ12}\right)$
 ${\mathrm{N12}}{:=}\left[{}\right]$ (2.5)

Example 2.

Find the normalizer for the Lie algebra of infinitesimal rotations in 3 dimensions.

 M > $\mathrm{DGsetup}\left(\left[x,y,z\right],N\right)$
 ${\mathrm{frame name: N}}$ (2.6)
 N > $\mathrm{Γ2}≔\mathrm{evalDG}\left(\left[z\mathrm{D_y}-y\mathrm{D_z},x\mathrm{D_y}-y\mathrm{D_x},z\mathrm{D_x}-x\mathrm{D_z}\right]\right)$
 ${\mathrm{Γ2}}{:=}\left[{\mathrm{D_y}}{}{z}{-}{\mathrm{D_z}}{}{y}{,}{-}{\mathrm{D_x}}{}{y}{+}{\mathrm{D_y}}{}{x}{,}{\mathrm{D_x}}{}{z}{-}{\mathrm{D_z}}{}{x}\right]$ (2.7)
 N > $\mathrm{N2}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{Γ2}\right)$
 ${\mathrm{N2}}{:=}\frac{{x}{}{\mathrm{_F1}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right){}\sqrt{{-}{{z}}^{{2}}}{}{\mathrm{D_x}}}{{z}}{+}\frac{{y}{}{\mathrm{_F1}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right){}\sqrt{{-}{{z}}^{{2}}}{}{\mathrm{D_y}}}{{z}}{+}{\mathrm{_F1}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right){}\sqrt{{-}{{z}}^{{2}}}{}{\mathrm{D_z}}$ (2.8)

Now let us find the normalizer for the Lie algebra of infinitesimal rotations in three dimensions within the infinite-dimensional Lie algebra of divergence-free vector fields.

First define a general vector field on with arbitrary coefficients  and .

 N > $U≔\left[A,B,C\right]\left(x,y,z\right)$
 ${U}{:=}\left[{A}{}\left({x}{,}{y}{,}{z}\right){,}{B}{}\left({x}{,}{y}{,}{z}\right){,}{C}{}\left({x}{,}{y}{,}{z}\right)\right]$ (2.9)
 N > $Z≔\mathrm{evalDG}\left(U\left[1\right]\mathrm{D_x}+U\left[2\right]\mathrm{D_y}+U\left[3\right]\mathrm{D_z}\right)$
 ${Z}{:=}{A}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{D_x}}{+}{B}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{D_y}}{+}{C}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{D_z}}$ (2.10)

We use the keyword argument auxiliaryequations to require that the vector field $Z$ be divergence-free.

 N > $E≔\left[\mathrm{diff}\left(A\left(x,y,z\right),x\right)+\mathrm{diff}\left(B\left(x,y,z\right),y\right)+\mathrm{diff}\left(C\left(x,y,z\right),z\right)=0\right]:$

 N > $\mathrm{N2div}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{Γ2},\mathrm{ansatz}=Z,\mathrm{unknowns}=U,\mathrm{auxiliaryequations}=E\right)$
 ${\mathrm{N2div}}{:=}\frac{{x}{}{\mathrm{_C1}}{}{\mathrm{D_x}}}{{\left({{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{3}{/}{2}}}{+}\frac{{y}{}{\mathrm{_C1}}{}{\mathrm{D_y}}}{{\left({{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{3}{/}{2}}}{+}\frac{{\mathrm{_C1}}{}{z}{}{\mathrm{D_z}}}{{\left({{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)}^{{3}{/}{2}}}$ (2.11)

Example 3.

In this example we shall calculate the normalizers for a Lie algebra of vector fields which depends upon a parameter $\mathrm{α}$.  We find that Nor(${\mathrm{Γ}}_{3}$) mod has dimension 2 for and dimension 3 for .

 M > $\mathrm{Γ3}≔\mathrm{evalDG}\left(\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z},\left(x+z\right)\mathrm{D_x}+\mathrm{\alpha }y\mathrm{D_y}+z\mathrm{D_z}\right]\right)$
 ${\mathrm{Γ3}}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}{,}\left({x}{+}{z}\right){}{\mathrm{D_x}}{+}{\mathrm{α}}{}{y}{}{\mathrm{D_y}}{+}{z}{}{\mathrm{D_z}}\right]$ (2.12)
 G > $\mathrm{N3}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{Γ3},\mathrm{parameters}=\left\{\mathrm{\alpha }\right\},\mathrm{output}="list"\right)$
 ${\mathrm{N3}}{:=}\left[{z}{}{\mathrm{D_x}}{,}{\mathrm{D_x}}{}{x}{+}{\mathrm{D_z}}{}{z}{,}{\mathrm{D_y}}\right]{,}\left[{z}{}{\mathrm{D_x}}{,}{\mathrm{D_x}}{}{x}{+}{\mathrm{D_z}}{}{z}\right]{,}\left[\left\{{\mathrm{α}}{=}{0}\right\}{,}\left\{{\mathrm{α}}{=}{\mathrm{α}}\right\}\right]$ (2.13)

Example 4.

We calculate the normalizer of the infinitesimal Euclidean group in the infinitesimal pseudo-group of all contact transformation on a 3- dimensional contact manifold $M$ with coordinates with contact form

 N > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.14)

Here is the standard Euclidean metric on the $\left[x,y\right]$ plane and the standard contact form on $M.$

 M > $g≔\mathrm{evalDG}\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)$
 ${g}{:=}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.15)
 M > $\mathrm{\omega }≔\mathrm{evalDG}\left(\mathrm{dy}-z\mathrm{dx}\right)$
 ${\mathrm{ω}}{:=}{-}{\mathrm{dx}}{}{z}{+}{\mathrm{dy}}$ (2.16)

We use the command InfinitesimalSymmetriesOfGeometricObjectFields to find the Lie algebra of vector fields which preserves the metric $g$ and the Pfaffian system generated by $\mathrm{ω}.$

 M > $\mathrm{Γ4}≔\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[g,\left[\mathrm{\omega }\right]\right],\mathrm{output}="list"\right)$
 ${\mathrm{Γ4}}{:=}\left[{-}{y}{}{\mathrm{D_x}}{+}{x}{}{\mathrm{D_y}}{+}\left({{z}}^{{2}}{+}{1}\right){}{\mathrm{D_z}}{,}{\mathrm{D_y}}{,}{\mathrm{D_x}}\right]$ (2.17)

We define an arbitrary vector field $Z$ on $M$ and again use the command InfinitesimalSymmetriesOfGeometricObjectFields, this time to find the partial differential equations which the coefficients of must satisfy in order that this vector field be an infinitesimal contact transformation.

 M > $Z≔\mathrm{evalDG}\left(\mathrm{A1}\left(x,y,z\right)\mathrm{D_x}+\mathrm{A2}\left(x,y,z\right)\mathrm{D_y}+\mathrm{A3}\left(x,y,z\right)\mathrm{D_z}\right)$
 ${Z}{:=}{\mathrm{A1}}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{D_x}}{+}{\mathrm{A2}}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{D_y}}{+}{\mathrm{A3}}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{D_z}}$ (2.18)
 M > $E≔\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[\left[\mathrm{\omega }\right]\right],\mathrm{ansatz}=Z,\mathrm{unknowns}=\left[\mathrm{A1},\mathrm{A2},\mathrm{A3}\right]\left(x,y,z\right),\mathrm{output}="pde"\right)$
 ${E}{:=}\left[{z}{}{\mathrm{_K111}}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{x}}{}{\mathrm{A2}}{}\left({x}{,}{y}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{A1}}{}\left({x}{,}{y}{,}{z}\right)\right){}{z}{-}{\mathrm{A3}}{}\left({x}{,}{y}{,}{z}\right){,}{-}{\mathrm{_K111}}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{y}}{}{\mathrm{A2}}{}\left({x}{,}{y}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{A1}}{}\left({x}{,}{y}{,}{z}\right)\right){}{z}{,}\frac{{\partial }}{{\partial }{z}}{}{\mathrm{A2}}{}\left({x}{,}{y}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{z}}{}{\mathrm{A1}}{}\left({x}{,}{y}{,}{z}\right)\right){}{z}{,}{0}\right]{,}\left[{\mathrm{A1}}{}\left({x}{,}{y}{,}{z}\right){,}{\mathrm{A2}}{}\left({x}{,}{y}{,}{z}\right){,}{\mathrm{A3}}{}\left({x}{,}{y}{,}{z}\right){,}{\mathrm{_K111}}{}\left({x}{,}{y}{,}{z}\right)\right]$ (2.19)

Note that the factor  is an additional unknown satisfying .   The sought after normalizer of in the infinitesimal pseudo-group of contact transformations can now be computed.

 M > $\mathrm{Nor4}≔\mathrm{InfinitesimalPseudoGroupNormalizer}\left(\mathrm{Γ4},\mathrm{ansatz}=Z,\mathrm{unknowns}=E\left[2\right],\mathrm{auxiliaryequations}=E\left[1\right],\mathrm{output}="list"\right)$
 ${\mathrm{Nor4}}{:=}\left[{\mathrm{D_x}}{}{x}{+}{\mathrm{D_y}}{}{y}{,}{-}\frac{{z}{}{\mathrm{D_x}}}{\sqrt{{{z}}^{{2}}{+}{1}}}{+}\frac{{\mathrm{D_y}}}{\sqrt{{{z}}^{{2}}{+}{1}}}\right]$ (2.20)

We can check this result by noting that [i] the vector fieldspreserve and [ii] the normalizer is a 5-dimensional Lie algebra which contains ${\mathrm{Γ}}_{4}$ as an ideal.

 M > $\mathrm{LieDerivative}\left(\mathrm{Nor4},\mathrm{\omega }\right)$
 $\left[{-}{\mathrm{dx}}{}{z}{+}{\mathrm{dy}}{,}{0}{}{\mathrm{dx}}\right]$ (2.21)
 M > $\mathrm{LD}≔\mathrm{LieAlgebraData}\left(\left[\mathrm{op}\left(\mathrm{Γ4}\right),\mathrm{op}\left(\mathrm{Nor4}\right)\right],\mathrm{alg}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}\right]$ (2.22)
 M > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: alg}}$ (2.23)
 alg > $\mathrm{Query}\left(\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right],"Ideal"\right)$
 ${\mathrm{true}}$ (2.24)